Brownian-motion ensembles of random matrix theory: A classification scheme and an integral transform method

Heat Transport and Majorana Fermions in a Superconducting Dot-Wire System: An Exact Solution

“…A powerful way to represent the probability distributions of the RMT ensembles is to employ a classification scheme based on matrix-valued Brownian motion ensembles [16]. For the BdG classes with TR symmetry (DIII and CI) we may define the following functions:…”

“…where the random variables 푖 are related to the transmission eigenvalues 푖 via a simple procedure described in [16]. For a ballistic cavity we may follow [24] and take 푖 = 푖 and write the joint distribution as the stationary solution of the corresponding Brownian motion.…”

“…method that effectively maps (8) onto a much simpler Fokker-Planck problem in an image space of smaller dimension. The multidimensional integral transform is defined as [16] (…”

“…The problem has thus been reduced to solving (4) with an initial condition given by (3), which is the joint probability distribution of transmission eigenvalues of an Andreev quantum dot. Remarkably, an exact analytical solution can be constructed using an integral transform method [16] which provides a complete description of the crossover of the thermal conductance moments as a function of the system's length, covering all transport regimes: ballistic, metallic, and insulating. For a related study of a normal dot-wire system (WD class) with broken TR symmetry, see [22].…”

“…We obtain, in the continuum limit of a quantum chain with TR symmetry, an exact description of the crossover in thermal conduction between a superconducting chaotic ballistic cavity (a quantum dot) and a disordered multichannel superconducting quantum wire. The calculations were guided by a recent classification scheme of RMT Brownian motion ensembles [16] and were performed by means of a multivariate integral transform method proposed in [16,17]. More specifically, we obtain exact expressions for the first three moments of the heat conductance of two classes of disordered superconducting quantum wires with time-reversal symmetry and with a crossover to a quantum dot in the small length limit.…”